Here are max-period examples for X := reverse(X+C) mod 2^w found by random trial (C & period in hexadecimal): w=3 C=7 period=8 (max possible, trials=1) w=5 C=3 period=20 (max possible, trials=1) w=5 C=5 period=20 (max possible, trials=4) w=5 C=1D period=20 (max possible, trials=2) w=7 C=5B period=80 (max possible, trials=1) w=7 C=7 period=80 (max possible, trials=3) w=7 C=79 period=80 (max possible, trials=4) w=9 C=17 period=200 (max possible, trials=2) w=9 C=10B period=200 (max possible, trials=4) w=9 C=1DB period=200 (max possible, trials=3) w=9 C=75 period=200 (max possible, trials=6) w=11 C=77B period=800 (max possible, trials=28) w=11 C=2D7 period=800 (max possible, trials=31) w=11 C=245 period=800 (max possible, trials=39) w=11 C=7D period=800 (max possible, trials=8) w=13 C=323 period=2000 (max possible, trials=74) w=13 C=1631 period=2000 (max possible, trials=39) w=13 C=1D73 period=2000 (max possible, trials=38) w=13 C=49 period=2000 (max possible, trials=55) w=15 C=65 period=8000 (max possible, trials=299) w=15 C=30A3 period=8000 (max possible, trials=30) w=15 C=31A7 period=8000 (max possible, trials=65) w=15 C=513 period=8000 (max possible, trials=77) w=17 C=6F1D period=20000 (max possible, trials=94) w=17 C=21A3 period=20000 (max possible, trials=205) w=19 C=1F64B period=80000 (max possible, trials=2691) w=21 C=62D8D period=200000 (max possible, trials=292) w=23 C=7D9DDB period=800000 (max possible, trials=2105) w=25 C=1E40F47 period=2000000 (max possible, trials=2443) w=27 C=69833D7 period=8000000 (max possible, trials=1678) w=29 C=14149C61 period=20000000 (max possible, trials=2461) w=31 C=45C4ABF7 period=80000000 (max possible, trials=3984)